Fubini theorem for anticipating stochastic integrals in Hilbert space

Abstract

Let (X,l,μ) be a measure space, letW be a cylindrical Hilbert-Wiener process, and letϕ be an anticipating integrable process-valued function onX. We prove, under natural assumptions onϕ, that there exists a measurable version Yx,x eX, of the anticipating integral ofϕ(x) such that the integral ∫x Yxμ(dx) is a version of the anticipating integral of ∫X ϕ(x)μ(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.